Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence. As a consequence, we derive necessary and sufficient conditions under which the this restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in a special case. Since then, this problem has been studied, in several other special cases, in many papers. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions

Empirical cumulative entropy as a new trace elements indicator to determine the relationship between algae-sediment pollution in the Persian Gulf, southern Iran

In this paper, the amount of 19 elements in three species of algae and associated sediment in the northern margin of the Persian Gulf was investigated. A sampling of algae was performed on the coast with a length of 5 km in each station and surface sediment was sampled at the same time in low and middle intertidal zones. The values of elements in the samples were measured by using an inductively coupled plasma mass spectrometry (ICP-MS) device. Then, the amount of bioaccumulation factor in algae tissue relative to sediment (biota-sediment accumulation factor, BSAF) was determined. The value of BSAF was compared with the empirical cumulative entropy (ECE). ECE is based on comparing the element information in algae with those in sediments. The results showed that BSAF was very closely related to the ECE factor so that significant correlations were obtained for algae species of P. gymnospora (ECE = 0.477 BSAF, R2: 0.967), H. hamulosa (ECE = 0.542 BSAF, R2: 0.979), and C. membranacea (ECE = 0.356 BSAF, R2: 0.976). The ECE values > 0.4 were similar to those obtained for BSAF > 1, exhibiting that the element accumulation in algae was higher than in sediments. Based on ECE, to determine the vanadium accumulation in the environment, the C. membranacea algae are more appropriate than H. hamulosa. Overall, the data showed that ECE is a good alternative to BSAF in estimating marine pollution